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Let's fix $x \in \mathcal{X}$ a finite discrete space . we consider $F_{x}(h)=\epsilon \sum_{y}exp(-\frac{\Delta(x,y)}{\epsilon})$ where $\Delta(x,y) = \phi(x)+ \psi(y) + h(x)*(y-x) -c(x,y)$ and $\epsilon$ is some fixed constant, phi and psi just 2 $L^{1}$ functions which do not matter here.

If we consider it's derivative with r.t h(x) we have $F'_{x}(h)=-\sum_{y} exp(-\frac{\Delta(x,y)}{\epsilon})(y-x))$ which i can't see why we can consider it strongly convex , i can consider it stricly convex at most but it doesnt give me the existence of a unique minimizer. I needed to apply Newton's algorithm to find a solution to $F'_{x}(h)=0$ for each $x \in \mathcal{X}$ but since the minimizer existence is not clear i cant really apply it.

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  • $\begingroup$ Without knowing $\phi$, $h$, and $c$ it is hard to even see that $F_x'$ is convex at all. By the way, strict convexity is sufficient for guaranteeing a unique minimizer (this is a classical exercise in convex analysis). Also, I'm not sure how you intend to apply Newton's method in a finite discrete place. Newton's method is an algorithm defined on a continuous space. Please elaborate. $\endgroup$
    – Zim
    Jun 14, 2020 at 13:59
  • $\begingroup$ I'm trying to solve an optimal martingale transportation problem , this form i use is based on a Sinkhorn algorithm where we add a martingale constraint. $\endgroup$
    – Kupoc
    Jun 15, 2020 at 15:29

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